Modulated wave solutions

L. Recke and S. Yanchuk. On inverse amplitude synchronized modulated wave solutions in time reversable systems, in preparation. 

The state = ( = 0 is a steady state of Equations (11) for all parameter values, and by the choice of the constant terms in expansions for / and g, it is linearly stable for all values of a and a2. For the full system (8) this state corresponds to v = iv = 0, p = po= constant. The trivial steady state coexists with rotating-wave and modulated-wave solutions discussed next. Hence it plays much the same role in the ODE model as the homogeneous steady state plays in excitable media. That is, in all excitable media, there is a homogeneous steady state (e.g. u — v = O the reaction-diffusion model), which is linearly stable for all parameter values, and which coexists with rotating waves and modulated waves when they exist. The trivial steady state in the ODE model has the same character as this homogeneous state in excitable media. 

Before we proceed in solving Equation (27.15), we give some explanations for Equation (27.17). If there were not the last term in Equation (27.15), the one with which comes from the nonlinear term in Equation (27.13), we would expect the solution in the form Eie " + cc instead of Equation (27.17). This would be a modulated wave with a carrier component e " and an envelope F. We will see later that the modulation factor F will be treated in a continuum limit whereas the carrier wave will not. In other words, the carrier component of the modulated wave includes the discreteness and the procedure is called semi-discrete approximation. 

Sinusoidal modulation of n and a Slanted gratings Two-wave solution (plane waves) Bragg or near-Bragg solutions TE (H-mode, p) or TM (E-mode, p) polarizations Transmission and reflection gratings Neglects second derivatives No boundary reflections Approximate boundary conditions (E only) Neglects higher order waves... 

Having thus introduced some notions about the unperturbed system, we come back to the system (4.4.3) with perturbation. In analogy to the previous section, the solution of slowly phase-modulated waves may be sought in the form... 

The stability analysis of the periodic roll solution [54] is performed by introducing a small perturbation 5A (q, t) of the amplitude Ar with a modulation wave-vector s in Eqs. 32 and 33,... 

Consider first the RW solutions as a function of a. In Figure 5 stable RW states are seen both at low and high values of a. These states are, in fact, on a single branch of solutions which is unstable at intermediate values of a, (where the stable modulated waves are observed). This branch of RW solutions, together with its stability, has been computed as described in Sections 3.1 and 3.2. The results are shown in Figure 7. 

Fig. 19 (a) The bifurcation diagram for spatially periodic solutions at a forcing wave number q = 0.5 and modulation amplitudes a = 0 dotted), a = 0.01 dashed), and a = 0.03 solid), (b)—(d) The three solutions yrs (x) over one period corresponding to the different branches of the bifurcation diagram for a = 0.03 and = 1. Figure from [122]. Reprinted with permission by Springer... 

Another example for the HMRRD electrode is given in Fig. 9 for Fe in alkaline solutions [12, 27]. The square wave modulation of the rotation frequency co causes the simultaneous oscillation of the analytical ring currents. They are caused by species of the bulk solution. Additional spikes refer to corrosion products dissolved at the Fe disc. This is a consequence of the change of the Nemst diffusion layer due to the changes of co. This pumping effect leads to transient analytical ring currents. Besides qualitative information, also quantitative information on soluble corrosion products may be obtained. The size of the spikes is proportional to the dissolution rate at the disc, as has been shown by a close relation of experimental results and calculations [28-30]. As seen in Fig. 7, soluble Fe(II) species are formed in the po-... 

In the AC impedance method, a modulation of potential causes a modulation of concentration at the interface. As a consequence there is a modulation of the concentration gradient and hence of the current density. The modulation of concentration decays with distance away from the interface. Thus there is a characteristic penetration depth into the solution phase of the concentration wave, SAC, where SAc = (2Dlw)m. The time scale for chemical processes, either in the solution or at the interface, probed by the perturbation, is simply / >. If the concentration boundary... 

To obtain the bulk properties of the crystal, the crystal is assumed infinite and the boundary condition for is periodicity within one arbitrary domain containing a large number of unit cells. This boundary condition gives rise to solutions (79) that are running waves modulated by a function Wkm(r) that has the periodicity of [/(r),... 

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